1. Technical Field
The invention relates to risk analysis. More particularly, the invention relates to an integrative method for modeling assets from many different asset classes. It is illustrated by application to the world equity market, when each country or regional equity market may be treated as an asset class.
2. Description of the Prior Art
All my fortunes are at sea.
My ventures are not in one bottom trusted, nor, to one place; nor is my whole estate upon the fortune of this present year.                William Shakespeare, The Merchant of Venice (1598)        
Shakespeare reminds us that the perils of investment have been with us always. Likewise, the insight that these risks can be reduced through diversification is ancient. A very recent idea, however, is that the risks can be measured and the degree of diversification can be optimized. This new insight rests on a technological advance—the risk model—which itself rests on the technological revolution produced by the computer and, more specifically, by the silicon chip.
Within the world of risk modeling there has, until recently, been a need to compromise on objectives. As long as one's focus was confined to securities within a single market, U.S. equities say, detailed analyses have been possible. When one's perspective broadens to consider equities from around the world, currencies, etc., the depth of analysis contracts. Fundamentally this compromise between depth and breadth has been necessitated by the limitations of computing technology. In the last few years, however, these limitations have been much reduced.
It would be desirable to harness the recent increase in computing power to achieve an analysis of global asset risk which is both broad and deep.
Fundamental Concepts
The Covariance Matrix
A standard approach to characterizing financial risk is to measure the variance of the return series. Let ri(t) be the return to asset i in period t and define the asset-to-asset covariance matrix Ω byΩij(t)=cov(ri(t),rj(t))where cov( ) is the covariance operator. Suppose h is the vector whose component hi gives the fraction of wealth invested in asset i by a particular portfolio. Then the variance of the portfolio return is given byhtΩh
More generally if h1 and h2 are vectors defining distinct portfolios, then the covariance of the returns to those portfolios is given byh1tΩh2This reduces the risk analysis of a portfolio to the problem of determining a good estimate {circumflex over (Ω)} of the asset-by-asset covariance matrix Ω.
In the last 25 years, there has been rapid growth in types of assets, such as options, where the variance may not be an adequate description of risk. In these cases, there is a set of underlying variables, such as underlying asset prices, whose riskiness is captured by their variances. Then Ω is the covariance matrix of the underlying variables, such as underlying asset prices. For expositional simplicity, and because the variance does provide a first measure of risk, we will use portfolio covariance as a risk measure and use Ω to denote the asset level covariance matrix. The simplest estimator {circumflex over (Ω)} is the historical covariance matrixΩijhist(t)=cov({ri(u)rj(u)}u=t,1)
An important practical point is that the number of periods t entering into this estimate is constrained by economic realities. Let T denote the total length of time over which returns are observed and let Δ denote the observation interval. Then t=T/Δ. In general T is limited by two circumstances. First, assets have finite lives. Second, the economy itself is evolving and this evolution limits the relevance of data from the distant past. It is desirable, though not essential, for T to be at least five years. For the risk horizons of interest in a portfolio management context, e.g. from one quarter to a few years, a rule of thumb is that the observation interval Δ is best set at a one month horizon. Taking T at five years and Δ at one month results in the number of periods t being 60. The choice of 60 is not a hard number, but it represents a reasonable and necessary compromise.
The statistical properties of the estimator {circumflex over (Ω)} depend crucially on two parameters: the number of periods t in the estimate and the number of assets, N, covered by the estimate. If N>t then we may find portfolios h such thatht{circumflex over (Ω)}h=0over the sample period. Such portfolios appear to be risk free, but in fact are not. Technically this condition is expressed by saying that {circumflex over (Ω)} fails to be positive definite. If a covariance matrix which is not positive definite is used for portfolio construction there will be a strong tendency to buy into the apparently risk free portfolios. The result is a severely biased risk estimate, with realized risks proving significantly higher than forecast risks. For this reason a positive definite covariance matrix is a basic requirement, and thus one requires t>N. Econometric considerations limit t to approximately 60, while practical applications may require N on the order of 1000. Thus, the historical asset-by-asset covariance matrix is of only limited practical utility.The Factor Model
The limitations of the historical covariance matrix motivate the search for a more robust estimator of the asset covariances. The standard solution is to invoke a factor model. A factor model is a linear model for asset returns such that             r      i        ⁡          (      t      )        =                    ∑                  j          =          i                m            ⁢                                    X            ij                    ⁡                      (            t            )                          ⁢                              f            j                    ⁡                      (            t            )                                +                  ɛ        i            ⁡              (        t        )            where Xij(t) is termed the exposure of asset i to factor j, fj(t) is termed the return to factor j and εj (t) is termed the specific return to asset i. The returns need not be linear in the factors, as in the case of options. Our interest is in developing a covariance matrix for factors across many asset classes and the linearity assumption does not impact the interest in any manner: Non-linear instrument may by valued directly given factor realization. For portfolio risk analysis, the linear approach is a widely-used first order approximation which greatly speeds up computations. It is further assumed that the factors fj capture all common sources of return between assets, or equivalently thatcov(fj(t),εi(t))=0for all factors fj(t) and specific returns εj (t) and thatcov(εi(t),εk(t))=0for distinct assets i and k. In this case with a bit of algebraic manipulation one can show Ω=XFXt+Δ whereFij(t)=cov(fi(t),fj(t)) and                                          Δ            ij                    ⁡                      (            t            )                          =                ⁢                  cov          ⁡                      (                                                            ɛ                  i                                ⁡                                  (                  t                  )                                            ,                                                ɛ                  j                                ⁡                                  (                  t                  )                                                      )                                                  =                ⁢                  {                                                                      var                  ⁡                                      (                                          ɛ                      i                                        )                                                                                                                    if                    ⁢                                                                                   ⁢                    i                                    =                  j                                                                                    0                                            otherwise                                                        F is the common factor covariance matrix, and Δ is the (diagonal) matrix of specific risk. We estimate the quantities F and Δ historically.
The statistical properties of this model may be quickly summarized, based on our analysis of the sample covariance matrix. Since all assets have some specific risk, Δ>0, this insures that{circumflex over (Ω)}fact=X{circumflex over (F)}histXt+Δhistwill be positive definite. The accuracy of the common factor risk forecasts {circumflex over (F)}hist depend on t in the average case and on the quality ratio t/m in the worst case, where m is the number of factors.
Another source of error in the factor model derives from error in the factor structure, for instance due to omission of an important factor. This source of error can be controlled for empirically. For instance, the importance of the smallest included factor gives an estimate of the likely importance of the largest missing factor, if a systematic process for constructing the factor model has been followed.
Factor Modeling Techniques
There are two basic approaches for generating factor models, which are termed the exploratory and confirmatory approaches, respectively. The exploratory approach assumes that the returns are generated by a factor model but that nothing is known about the factor modelr=Xf+εVarious statistical techniques can then be applied to simultaneously estimate Xi and f(t) from the data ri(t). In this method X captures all the cross-sectional variation in i and f(t) captures all the temporal variation. Even so, however, X and f(t) are not uniquely defined, but rather are determined only up to a rotation of the exposures. Thus, the factors extracted by this technique are not directly interpretable. Interpretability of factors is an important consideration if a risk model is to be used for active portfolio management. In active management risks are deliberately taken in the effort to earn compensatory return. Thus, judgments must be formed as to whether or not one is willing to take on risk along a particular dimension. If the dimension is a statistical construct without an economic interpretation, there is little basis from which to form such judgments. It is usual in exploratory factor analysis to apply a rotation to the extracted factors in the hopes of arriving at an interpretation of the factors. The value of this interpretation, however, relies entirely on the analyst's judgment.
The confirmatory factor approach assumes that a priori information is available about the factor structure. In the returns based approach one assumes that the factor returns fj(t) are known. Then the exposures Xij(t) are found by regressing the asset returns on the factor returns. For instance, the market model assumes a single factor return, namely the market return m(t) in excess of the risk free rate r0(t) and it determines an asset exposure, the historical beta βhist by regressing asset returns in excess of the risk free rate on the market excess return
 ri(t)−ro(t)=αi+βihist[m(t)−r0]+εi(t)
A limitation of the returns based approach is that the estimated factor exposures may not be very interpretable. A variant, known as style analysis, attempts to correct this defect by carrying out a least squares estimation in which the exposures are restricted to lie in an a priori reasonable range. For instance, restricting the exposures to lie between 0 and 1 and to sum to 1 allows them to be interpreted as weights which describe how the factor returns are mixed together to best approximate the asset return. Note that if the data do not conform to the imposed restrictions then in general the condition.cov(fi(t),εj(t))=0will fail to hold, so style analysis cannot simultaneously guarantee interpretable exposures and a consistent factor structure.
In contrast to the returns based analysis, the exposure based approach to confirmatory factor analysis assumes that the exposure matrix X is known a priori. The factor returns f(t) are then estimated by regressing the asset returns on the exposures. The exposure approach differs from other factor modeling methods in that                1) the interpretive structure is unambiguous; and        2) the exposure matrix can more readily vary dynamically through time.        
However one generates a factor structure, one is faced with the problem of assessing its adequacy. For exploratory techniques one is guaranteed to find a structure that meets the basic assumption (1) and (2) of the factor model over the time period in which the model is estimated. The essential assessment then is whether the factor exposures estimated in this way remain stable in subsequent time periods. For returns based confirmatory factor analysis, stability of the factor exposures is again the basic criteria of success. For confirmatory analysis, by contrast, the technique only guarantees only that property (1) will hold. Thus, a test of the model is verifying how well property (2) holds—i.e. are the specific returns of distinct assets uncorrelated within measurement error? It is an advantage of the confirmatory methodology that this check of the model's adequacy may be made on the estimation data itself. By contrast, the exploratory method requires data subsequent to the estimation data to arrive before the internal consistency of the model can be assessed.
Cross Asset Class and Single Asset Class Modeling
When analyzing assets in a single asset class, the most accurate risk predictions are obtained by identifying the factors appropriate to that asset class and estimating a corresponding factor model. The number of factors and length of factor realization history is almost certain to be different among the various classes. In addition, the precision of the modeling is likely to vary by asset class as is the richness of the factor structure. Each asset class is likely to have different accessibility of individual asset data so that varying levels of granularity imply that certain items of information may be available within one class but not available within another.
Confirmatory factor analysis for equities in a single country reveals that equity returns are driven by industry and style related factors. Industry related factors are self-explanatory: each firm functions within a single industry or across multiple industries, and the exposure of a firm to different industries may be computed by using a combination of sales, assets, and income from the different industries. Style related factors are based on firm fundamentals, such as size, growth, or relative trading activity, and exposure to style based factors are computed using fundamental accounting information, e.g assets, or market information, e.g. capitalization, trading volume. The prevalence of different industries, the availability of fundamental and market data, and the local behavior of the market then determine the final factor structure that is used for equities in a single country.
For fixed income securities, the set of factors are default free (sovereign) or high grade (swap) interest rates and spreads for lower grade instruments. The first set of factors is generally described by a set of zero coupon yields, or by the first three principle components of the set of zero coupon yields. Depending on the availability of data, the spread factors can be divided into sector and rating related spreads. In the U.S., for example, it is possible to get a fairly rich set of spread factors because of a long history of data, whereas such a history may not be available for other markets.
When integrating across many assets classes, it has in the past been difficult to work with the factors from the single asset class models, particularly for equities. The problem, discussed in detail below, is simply that there are too many factors, differing factor histories, and different types of statistical analyses that may be applied to each asset class. The natural simplification, which eliminates the need for aggregating different types of model, is to impose a common factor model structure across all asset classes. For example, one could impose a single factor model so that the aggregation process requires only the estimation of the correlation between the various factors, while at the individual asset level one has only to estimate asset exposures and residual risk. As described below in detail, this has been past practice when modeling global equities. This greatly simplifies the aggregation task but either leaves the portfolio manager with an inferior model or yields inconsistent results between the various levels within the firm hierarchy, neither of which result is necessary, or desirable.
To motivate our solution to the cross-asset class problem, we next illustrate the natural tension that arises between cross and within asset class risk analysis by focusing our attention on the details of modeling of global and single country equities. We then show how the invention solves the tension by providing an approach that integrates single asset class models into an aggregate asset class model. Again, this resolution is illustrated by a detailed application to the global equity market.
Prior BARRA Research
The existing approach to global equity factor modeling is closely based on the Grinold, Rudd, and Stefek model of 1989. In this approach, local market residual returns{tilde over (r)}i(t)=ri(t)−rfi(t)−βi(t)mi(t)are calculated, where                ri(t) is the return to asset i in local currency        rfi(t) is the risk free rate for the local currency of asset i        βi(t) is the beta from a five-year CAPM regression        mi(t) is the local capitalization weighted equity index return        
The market residual returns are then fit to an exposure based factor model of the form                     r        ~            t        ⁡          (      t      )        =                    ∑                  j          =          1                          N          i                    ⁢                                    X            ij            1                    ⁡                      (            t            )                          ⁢                              f            j            1                    ⁡                      (            t            )                                +                  ∑                  k          =          1                4            ⁢                                    X            ik            2                    ⁡                      (            t            )                          ⁢                  f          k          2                      +                  ɛ        i            ⁡              (        t        )            
Here the Xij1(t) are Ni industry exposures where for every asset i there is a unique j such thatXij1(t)=1 and for all k≠jXik1(t)=0
The quantities fj1(t) are interpreted as returns to globally defined industries. In addition, Barra's global equity model contains a set of style factors. These are embodied in Xik2(t), which are four statistical characterizations of asset i, termed SIZE, SUCCESS, VALUE and VARIABILITY IN MARKETS. These measures are based on asset characteristics normalized against the local market. The estimation of the Barra model is carried out over approximately 2000 assets, drawn from some 25 developed markets. The exact composition of the estimation universe varies through time. The commercial version of this model is known as the Global Equity Model. It exists in two different versions distinguished by slightly different industry classification schemes and estimation universes.
Since the original formulation of the Global Equity Model we have learned much about global equities:    1) The power of exposure based factor analysis has been confirmed. Exploratory factor analysis and returns based analysis have generally confirmed findings first achieved with the Global Equity Model without generating fundamentally new insights themselves (Drummen and Zimmerman 1992, Heston and Rouwenhorst 1995). At the same time the limitations of returns based analysis have been convincingly documented (Hui 1996).    2) All studies have found country factors to be important. However, the most recent studies find country factors to be of declining importance in Europe (Becker, Connor and Curds 1996, Freimann 1998, Connor and Herbert 1998). In discussions about the relative importance of country and global industry factors it is accordingly useful to distinguish the European situation from the rest of the world.    3) The Global Equity Model found global industries to be important explanatory concepts. However, other studies suggest that this factor structure may be partially an analyst construct rather than an organic feature of asset returns. From a principal component analysis only the energy sector emerges as a clearly global industry (Hui 1995). Regional and country industries are found to carry substantially more explanatory power than global industries (Stefek 1991, Becker, Connor and Curds 1996). Just within Europe, continental and UK industries are found to be separate concepts (Connor and Herbert 1998).    4) The importance of different factors fluctuates through time (Rouwenhorst 1999). Recently industry and size factors have been of considerable importance (Chandrashekaran, Hui and Rudd 1999; Cavaglia, Brightman and Aleed 2000; Bach, Garbe and Weiss 2000).    5) A number of studies have emphasized that global equities exhibit considerable inhomogeneities. Simple differences in available data handicap a number of studies, e.g. Chaumeton and Coldiron 1999, Chandrashekaran, Hui and Rudd 1999. Specific risks are best understood in a local context (Grinold and Drach 1992). Local markets exhibit distinct behaviors of market risk (Hui 1997). Pricing appears to be local (Hui 1994). Regional concepts appear to be valid in some but not all parts of the world (Hui 1996). Local market factor structures differ (Hui 1997). Attempts to define a subuniverse of truly global companies appear problematic (Chaumeton and Coldiron 1999, Chandrashekaran, Hui and Rudd 1999). The UK appears to be only partially integrated into Europe (Connors and Herbert 1998).    6) Risk indices are found to have high marginal explanatory power per factor (Hui 1995). However they have been entirely ignored or relatively underutilized in most studies. Compared to single country factor models, fundamental, i.e. accounting, data are underutilized in the global setting. The principal difficulty is variations in disclosure and accounting standards around the globe.
Synthesizing all of this information, we are led to a new vision of global equities. Whereas the Global Equity Model saw global equities as a homogeneous group caught in a simple factor structure, we now see each local market as the homogeneous grouping with different markets linked together into a global matrix by various regional and global effects. The natural realization of this vision is to fit a factor model to each local market. The local models can be customized to each market to capture its special features and to best exploit the available data. The local analysis must then somehow be integrated into a global analysis. The work of Hui has been pointing in this direction since 1995. How to achieve the integration of local models has, however, been an elusive point. It would be advantageous to resolve this difficulty.